Divine proportion - abstract. Divine proportion - abstract The mysterious world of proportions

Class: 6

Presentation for the lesson

Back forward

Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.

Lesson type: generalization lesson

Equipment: computer, interactive whiteboard.

Lesson objectives:

Educational:

generalization and systematization of students’ knowledge on this topic;
strengthening the applied and practical orientation of the studied topic;
establishing intra-subject and inter-subject connections with other topics in the course of mathematics, geography, physics, astronomy, biology, chemistry.

Educational:

broadening the horizons of students,
vocabulary replenishment;

Educational:

nurturing interest in the subject and related disciplines,
to cultivate a sense of beauty, a sense of patriotism.

I. Organizational moment:

1) message about the topic of the lesson (slide 1);

2) communicating the goals and objectives of the lesson.

II. Updating knowledge on the topic “Proportions”:

What is the ratio of two numbers called?
What does the ratio of two numbers show?
What is proportion?
What are the terms of this proportion called?
What basic property do the members of a proportion have?
Which two quantities are called directly proportional? (give examples of directly proportional quantities).
Which two quantities are called inversely proportional? (examples).

III. From the history of proportion. (slides 2-5)

Word "proportion" comes from the Latin word proportionio, meaning proportionality, a certain relationship between parts. Proportions have been used since ancient times to solve various problems in mathematics.

Even in ancient Greece, mathematicians used such a device as PROPORTION.

Proportion is the equality of the ratios of two or more pairs of numbers or quantities.

In Babylon, plans of ancient cities were drawn using proportions. The drawing shows a plan of the ancient Babylonian city of Nippur found during excavations. When scientists compared the results of excavations of the city with this plan, it turned out that it was made with great accuracy.

IV. Practical application of proportions. (slide 6-7)

Mathematics is used in almost all areas of human life. And in everyday life we use mathematical skills, including proportion.

1. Architecture (slides 8-11)

When building a temple in honor of the goddess Diana, the Romans took the proportion that distinguishes slender women: the thickness of the column was only 1/8 of its height. Thanks to this, the columns seemed taller than they actually were, precisely due to the reduction in thickness. The architecture included both types of columns, one maintaining male and the other female proportions in the relationship between base and height.

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole.

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamen indicate that Egyptian craftsmen used the ratios of the golden division when creating them.

Solve problems.

1. 4 thousand bricks are used to build a house. How many thousand bricks are needed to build 15 similar houses?

2. To transport sand during construction, 14 vehicles with a carrying capacity of 4.5 tons were required. How many vehicles with a carrying capacity of 7 tons would be required to transport the same sand?

2. Cooking (slides 12-13)

The concept of proportion is used in cooking. When we prepare a dish, we try to use the amount of ingredients indicated in the cookbook. This is done so as not to spoil the dish. If we take more salt, we will oversalt it, and if we take less, it will not be tasty. Another proportion allows you to calculate the amount of ingredients for preparing the same dish for a different number of guests.

Solve problems

3. To make jam from 2 kg of gooseberries, you need 3 kg of sugar. How many kg of sugar are needed to make jam from 4.4 kg of gooseberries.

4. During drying, the mass of apples changed from 20 kg to 18.2 kg. By what % did the mass of apples decrease during drying?

3. Medicine (slides 14-16)

In medical practice, doctors monitor how much and when to give medicine to the patient. In the right doses, the medicine gives a therapeutic effect, in smaller doses it is useless, and in large doses it is harmful. When making medicines, proportions are also observed. Accuracy is necessary here, since if the proportions of the ingredients that make up the medicine are violated, the result may be not medicine, but poison. Ratios and proportions are also used in pharmacies in the manufacture of medicines and medicinal drinks. To make a drug, you need to know exactly how many parts there are for any part.

Solve problems

5. For a medicinal chamomile decoction, 20 g of dry chamomile are needed per 100 g of boiling water. How many g of chamomile are needed for 500 g of decoction?

6. The patient is prescribed a course of medication, which must be taken 250 mg twice a day for 7 days. One package of the medicine contains 10 tablets of 125 mg. What is the smallest number of packages needed for the entire course of treatment?

4. Chemistry (slides 17-19)

The theory of proportions has occupied a well-deserved place in solving problems in chemistry.

For example. What is the percentage concentration of the solution obtained by dissolving 5 g of table salt in 45 g of water?

Solve problems

7. 100 g of salt was dissolved in 2.4 liters of water. What is the concentration of the resulting solution?

8. There are 90 g of 80% vinegar essence. What is the largest amount of 9% table vinegar that can be obtained from it?

5. Technology (slides 20-23)

In technology lessons we also use proportion. When we want to sew something smaller or larger, we reduce or increase the pattern to the size we need. For example, a pattern for an apron for yourself and a doll. The sizes of the elements of the doll's apron differ from the corresponding sizes of my apron by the same number of times.

Solve problems

9. The overcasting machine processes 0.6 m of fabric in 2.16 minutes. How many meters can you sweep in 1.44 minutes?

10. It takes 1.2 m to make a children’s dress. How much fabric is needed for a dress for adults if the cost is 40% more?

6. Physics (slides 24-25)

Since ancient times, people have used various levers. Paddle, crowbar, scales, scissors, swing, wheelbarrow, etc. - examples of levers. The gain that the lever gives in the applied effort is determined by the proportion, where M and m are the masses of the loads, and L and l are the “arms” of the lever.

Solve problems

11. Using the lever rule, find M if l=2 m, L=8 m, m=4 kg.

12. In the city of Zhukovsky, demonstration flights of aircraft take place at the MAKS air show. A fighter aircraft such as the MIG-29 requires about 7.5 tons of kerosene for 3 hours of flight. How many tons of kerosene will the MIG-29 need for 7 hours of flight?

7. Modeling (slides 26-27)

Solve problems

13. The length of the car model is 42 cm. What is the length of the car if its dimensions are reduced by 10,000 times.

14. The sailboat model uses 60 cm of fabric. How many meters of fabric are needed to make three similar sailboats?

8. Geography. (slides 28-30)

In geography, proportion is also used - scale . Scale is the ratio of the length of a segment on a map or plan to the length of the corresponding segment on the ground. The scale shows how many times the distance on the plan is less than the indicated distance in reality.

Solve problems

15. Find the distance from Moscow to the North Pole, if on the map this distance is 3.5 cm, and M is 1:100000000.

16. Find the distance on the map between the cities of Rostov-on-Don and Moscow, if the distance between them is 1200 km, and M is 1:50000000.

V. Student reports on the application of proportion.

9. Fine arts. (slides 30-37)

10. Biology.(slides 38-39)

11. Music (slides 40-41)

12. Literature (slides 42-44)

VI. Conclusion (slide 45)

Since ancient times, people have used mathematics in everyday life. One of them is proportion. It is used from cooking to works of art such as sculpture, painting, architecture, and also in wildlife.

VII. Homework.

Literature

From the experience of conducting extracurricular work in mathematics in high school. Sat. articles edited by P. Stratilatova. – M.: Uchpedgiz, 1955.
D.Pidow. Geometry and art. – M.: Mir, 1989.
Magazine “Quantum”, 1973, No. 8.
Magazine “Mathematics at School”, 1994, No. 2, No. 3.
G. Mishkevich “Doctor of Entertaining Sciences” - M.: Knowledge, 1986
I. Ageeva “Entertaining materials on computer science and mathematics” – M.: Creative Center, 2005.
CD-ROM “From Plow to Laser 2.0”, New disc, 1998
Standard basic software package for educational institutions First Aid 1.0 Disc No. 56 New generation electronic educational resources Disc 1/1 DVD
http://www.sak.ru/reference/famous-buildings/famous-building5-1f.html Parthenon
http://www.foxdesign.ru/legend/apollo1.html Apollo Belvedere
http://www.sunhome.ru/journal/184 Mona Lisa
http://www.beseder.co.il/image-gallery/11897/1/1/ Leonardo da Vinci

At school, in the lessons of natural sciences: physics, chemistry, biology, astronomy, geography and in the lessons of the humanities: history, literature, native and foreign languages, we study nature and society. In music, drawing, drawing, and gymnastics lessons, we are introduced to the world of art. In addition to these disciplines, these subjects, throughout all school years we study mathematics: arithmetic, algebra, geometry, trigonometry. What sciences should these disciplines be classified as? What is the subject of their study? Many scientists classify mathematics as a natural science, since mathematics studies the world around us: objects and phenomena of nature, society and human thinking. Physics, chemistry, biology study objects and phenomena of the world around us from the aspect of their quality. Mathematics studies the same objects, phenomena from the side of their quantity, space and time, they say - from the side of their form.

Therefore, scientists consider mathematics to be a natural science that studies our material world. Mathematics permeates all branches of knowledge, including the humanities. Nowadays economics, philology and other sciences cannot do without mathematics. Therefore, some scientists consider mathematics to be a layer between the natural sciences and the humanities.

The great German mathematician Carl Friedrich Gauss once called mathematics “the queen of all sciences” and “the queen and servant of all sciences.” This is what she is called for her noble service to almost all sciences.

There are many methods in mathematics that allow you to solve certain problems. Even in ancient Greece, mathematicians used such a device as PROPORTION.

Proportion is the equality of the ratios of two or more pairs of numbers or quantities. For example, the dimensions of a model of a machine or structure differ from the dimensions of the original by the same factor, which specifies the scale of the model. Therefore, if you select 4 points A, B, C and D on the original and designate the corresponding points on the model as A1, B1, C1 and D1, then the equality == will hold. This equality of relations is called proportion. It shows that the ratio of distances between points on the original is the same as the ratio of distances between corresponding points on the model.

In ancient times, the idea of proportionality was used in an implicit form when solving problems using the complex position method: they gave the desired quantity a value, calculated what value one of these quantities should have, and compared it with the condition of the problem. The ratio of the values gave the coefficient by which the selected value must be multiplied to obtain the correct answer.

Proportions began to be studied systematically in Ancient Greece. At first, only proportions made up of natural numbers were considered, and therefore it was believed that the numbers a, b, c, d form a proportion if a is the same multiple, the same fraction or the same fraction of b as c of d. In the 4th century. BC e. The ancient Greek mathematician Eudoxus gave a definition of proportion, composed of quantities of any nature. Ancient Greek mathematicians solved problems that are solved today using equations, and algebraic transformations were replaced by the transition from one proportion to another.

In modern mathematics, various PROPERTIES OF PROPORTIONS are used.

The main property of proportion. If a: b = c: d, then a∙d = b∙c

Reversal of proportion. If a: b = c: d, then b: a = d: c

Rearrangement of middle and extreme terms. If a: b = c: d, then a: c = b: d (rearrangement of the middle terms of the proportion), d: b = c: a (rearrangement of the extreme terms of the proportion).

Increasing and decreasing proportions. If a: b = c: d, then

(a + b) : b = (c + d) : d (increase in proportion),

(a – b) : b = (c – d) : d (decreasing proportion).

Making proportions by adding and subtracting. If a: b = c: d, then

(a + c) : (b + d) = a: b = c: d (composing proportions by addition),

(a – c) : (b – d) = a: b = c: d (composing proportions by subtraction)

Mathematics is used in almost all areas of human life. And in everyday life we use mathematical skills, including proportion.

COOKING

MEDICINE

TECHNOLOGY

GEOGRAPHY

In geography, proportion is also used - scale. Scale is the ratio of the length of a segment on a map or plan to the length of the corresponding segment on the ground. The scale shows how many times the distance on the plan is less than the indicated distance in reality.

There are different types of scale: numerical, linear and named. The numerical scale is written as a fraction, the numerator of which is one, and the denominator is the degree of reduction in the projection. For example, a scale of 1:5,000 shows that 1 cm on the plan corresponds to 5,000 cm (50 m) on the ground. The larger scale is the one whose denominator is smaller. For example, a scale of 1:1,000 is larger than a scale of 1:25,000. The numerical scale is used to determine how many times all distances on the plan are reduced. The larger the number in the denominator of the fraction, the greater the number of times the actual distance is reduced, the smaller the map.

The entry “1 cm - 10 m” is called a named scale, and the distance on the ground corresponding to 1 cm on the plan is called the scale value. Using the scale value is very convenient for determining distances.

A linear scale is also placed on the plans. Linear scale is a graphic scale in the form of a scale bar divided into equal parts. This is a straight line divided into equal parts (usually centimeters). At each division of the line, the distance on the ground corresponding to it is indicated. The first division to the left of 0 is divided into smaller parts. Using a linear scale, you can find out the exact sizes of objects depicted on the terrain plan and the distances between them.

Task. Find the distance from Moscow to the North Pole, if on the map this distance is 3.5 cm, and M is 1:100000000.

Let's make a proportion: x=, i.e. x= 350000000cm=3500km.

Answer. The distance on the ground from Moscow to the North Pole is 3500 km.

ART

Alexey Petrovich Stakhov, Doctor of Technical Sciences (1972), Professor (1974), Academician of the Academy of Engineering Sciences of Ukraine writes about harmony:

“For a long time, people have been striving to surround themselves with beautiful things. Already the household items of the inhabitants of antiquity, which, it would seem, pursued a purely utilitarian purpose - to serve as a storage place for water, a weapon for hunting, etc., demonstrate man’s desire for beauty. At a certain stage of his life development, man began to ask the question: why is this or that object beautiful and what is the basis of beauty? Already in Ancient Greece, the study of the essence of beauty, beauty, formed into an independent branch of science - aesthetics, which among ancient philosophers was inseparable from cosmology. At the same time, the idea was born that the basis of beauty is harmony.

Beauty and harmony have become the most important categories of knowledge, to a certain extent even its goal, because ultimately the artist seeks truth in beauty, and the scientist seeks beauty in truth. The beauty of a sculpture, the beauty of a temple, the beauty of a painting, a symphony, a poem. What do they have in common? Is it possible to compare the beauty of the temple with the beauty of the nocturne? It turns out that it is possible if common criteria for beauty are found, if general formulas of beauty are discovered that unite the concept of beauty of a wide variety of objects - from a daisy flower to the beauty of a naked human body? ".

The famous Italian architectural theorist Leon Battista Alberti, who wrote many books on architecture, said the following about harmony:

“There is something more, made up of the combination and connection of three things (number, limitation and placement), something with which the whole face of beauty is miraculously illuminated. We call this harmony, which, without a doubt, is the source of all charm and beauty. After all, the purpose and goal of harmony - to arrange parts, generally speaking, different in nature, by some perfect relationship so that they correspond to one another, creating beauty. It embraces the whole of human life, permeates the entire nature of things. For everything that nature produces is all measured by the law of harmony. And "Nature has no greater concern than that what she produces is perfect. This cannot be achieved without harmony, for without it the highest harmony of the parts disintegrates."

The Great Soviet Encyclopedia gives the following definition of the concept of “harmony”:

“Harmony is the proportionality of parts and the whole, the merging of various components of an object into a single organic whole. In harmony, internal orderliness and measure of being are externally revealed.”

The “golden proportion” is a mathematical concept and its study is primarily a task for science. But it is also a criterion of harmony and beauty, and this is already a category of art and aesthetics, which studies harmony and beauty from a mathematical point of view.

In the classics of fine art, for many centuries, a technique for constructing proportions has been traced, called the golden ratio, or the golden number. (this term was introduced by Leonardo da Vinci). The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole.

a: b = b: c or c: b = b: a.

In art, the number 1:1.62 or

That is, an approximate expression of the ratio of a smaller value in proportion to its larger value.

The golden number is observed in the proportions of a harmoniously developed person: the length of the head divides the distance from the waist to the top of the head in the golden ratio.

In addition, there are several more basic golden proportions of our body: the distance from the fingertips to the wrist and from the wrist to the elbow is 1:1. 618 the distance from shoulder level to the top of the head and the size of the head is 1:1. 618 the distance from the navel point to the top of the head and from shoulder level to the top of the head is 1:1. 618 the distance of the navel point to the knees and from the knees to the feet is 1:1. 618 the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1:1. 618 the distance from the tip of the chin to the top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1. 618 the distance from the tip of the chin to the top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1. 618

In works of fine art, artists and sculptors, consciously or subconsciously, trusting their trained eye, often use the ratio of sizes in the golden ratio.

The same phenomenon is observed in other structures of nature: in the spirals of mollusks, in the corollas of flowers and in many other things familiar to us, for example, the arrangement of leaves on a shoot also obeys the golden number!

Since ancient times, people have used mathematics in everyday life. One of them is proportion. It is used from cooking to works of art such as sculpture, painting, architecture, and also in wildlife.

Today we will get acquainted with an unusual proportion called the golden ratio and even the divine proportion. You will learn what role this proportion plays in the world around you, how it is related to the concept of harmony and how and why it is used in art (painting, architecture, photography...), design...

In painting, photography, and design, the golden ratio is very often used in the form of classical composition techniques, which you can read about by looking at any website dedicated to these types of art.] The main recommendation is as follows. The object, which is the central figure in the composition, does not always have to be located in the center. Certain points in the composition automatically attract attention. There are 4 such points, and they are located at a distance of 3/8 and 5/8 from the edges of the picture. Having drawn a grid, we get these points at the intersections of the lines (see photo).

The golden ratio refers to such a proportional division of a segment into unequal parts. In which the length of the entire segment is related to its larger part, as the length of the larger part is related to the length of the smaller one. This ratio is equal to the irrational number Ф= The golden ratio is first found in Euclid’s Elements (300 BC). Luca Pacioli, a contemporary of Leonard da Vinci, called it “divine proportion.” The golden ratio is denoted by the symbols PHI or Ф (in honor of the ancient Greek sculptor Phidias, who always used the golden ratio in his works). The mathematician Fibonacci first obtained a sequence of numbers, named after him Fibonacci numbers 1,1,2,3,5,8,13,21,34,55 ... The peculiarity of this number series is that each of its terms, starting from the third, is equal the sum of the previous two: 1+1=2; 1+2=3; 2+3=5; 3+5=8 ...In this case, the ratio of two neighboring terms is equal to the golden ratio, i.e. number F. When considering patterns associated with the manifestation of the golden ratio, they usually use the reciprocal of the number F: 1/1.618 = 0.618 a+ba+b a bb: a = (a+b) : b

Question: What is common in the arrangement of polypeptide chains of nucleic acids, rose petals, mollusk shells, mammal horns, sunflowers, and distant cosmic galaxies? Answer: their structure is based on a golden (logarithmic) spiral. This spiral fits into a golden rectangle (the ratio of its length and width is equal to the number Ф). By successively cutting off squares from it and inscribing a quarter of a circle into each of them, we get a golden spiral (see photo). The role of the spiral in the structure of animal and plant objects was discovered by T. Cook, who proved that the phenomenon of growth is associated with the golden spiral. The carrier of the genetic code - the DNA molecule - consists of two intertwined helices. Not long ago, spiral structures were discovered in inanimate nature.

Phyllotaxis is a peculiar lattice arrangement of leaves, seeds, and scales of many plant species. The rows of nearest neighbors in such lattices unfold in spirals or twist in helical lines around a cylinder. Sunflower seeds are arranged in logarithmic spirals. In this case, the ratio of the number of left and right spirals is equal to the ratio of neighboring Fibonacci numbers. You can find sunflowers with a ratio of the number of spirals of 34/55 and 55/89.

The golden ratio permeates the entire history of art: the pyramids of Cheops, the famous Greek temple of the Parthenon, most Greek sculpture monuments, the unsurpassed Mona Lisa by Leonard da Vinci, paintings by Raphael, Shishkin, etudes by Chopin, music by Beethoven, Tchaikovsky, poems by Pushkin... this is not a complete list of outstanding works of art , filled with wonderful harmony based on the golden ratio. The photograph shows buildings in which the golden ratio was used to divide the main masses of their structures. It is usually believed that such division is used in buildings built in the classical style. However, look at the Smolny Cathedral, built in the Baroque style, and you will easily discover the golden ratio.

An ideal, perfect body is considered to have proportions equal to the golden ratio. The basic proportions were determined by Leonardo da Vinci, and artists began to consciously use them. The main division of the human body is the navel point. The ratio of the distance from the navel to the foot to the distance from the navel to the crown is the golden ratio. The ideal female figure is considered to be that of Aphrodite de Milo (see picture). Interestingly, the statistically average body sizes of various people are also subject to the rule of the golden section (this is evidenced by the anthropological studies of Zeising (1855), who measured almost 2000 people. Out of curiosity, you can check for yourself how close your body is to the ideal. Go to the Internet, type “ideal proportions of the human body”, take measurements and draw conclusions.There are certain rules by which the human figure is depicted, based on the concept of proportionality of the sizes of various parts of the body.

The shape of bird eggs is described by the golden ratio. Today it has already been established that with this configuration the strength characteristics of the shell are the highest. The perfect shape of a dragonfly's body is created according to the laws of the golden ratio: the ratio of the length of the tail and the body is equal to the ratio of the total length to the length of the tail. Summary For centuries, scientists have been using the unique mathematical properties of the golden ratio. This relationship is found in all living organisms, plants at all levels of their development. The universality of its manifestation in the structure of organs, systems, and their functional parameters suggests that it plays the role of a brick in the foundation of all life on Earth. Recent research in the field of astronomy and physics shows that this section is related to the entire Universe.

1. Divide a segment 16 cm long in relation to the golden ratio. Use Fibonacci numbers Option 1 – 3 and 5 Option 2 – 2 and 3 2. The length of the rectangle is 20 cm (Option 1), 15 cm (Option 2). Find the width of the rectangle such that the ratio of length to width is the golden ratio Ф = 1.6 Solve the problem by composing equation 3. Check how ideal one of the ratios of your palm is: the ratio of the length of the index finger to the length of its two phalanges from the end of the finger. Measure the indicated lengths using a ruler and find their ratio. Round the resulting number to tenths and compare with Ф=1.6 (determine how much more or less it is than the number Ф)

Description of the presentation by individual slides:

1 slide

Slide description:

2 slide

Slide description:

Introduction Representatives of many professions have to solve practical problems involving proportions. Artists, scientists, fashion designers, and designers make their calculations, drawings or sketches based on the Golden Ratio ratio. They use measurements from the human body, which was also created according to the principle of the “Golden Section”. We use math skills in everyday life, including proportion. Many problems in physics, chemistry, geography, etc. cannot be done without it. Hypothesis: a person in his activity is constantly faced with solving practical problems involving proportions.

3 slide

Slide description:

From the history of the study of proportions Proportions began to be studied in ancient times. In the 4th century BC. The ancient Greek mathematician Eudoxus gave a definition of proportion, composed of quantities of any nature. Ideas about beauty, order and harmony, and consonant chords in music were associated with proportions. The word “proportion” was coined by Cicero in the 1st century BC, which literally meant analogy, ratio.

4 slide

Slide description:

The theory of relations and proportions was expounded in detail in Euclid’s Elements (III century BC), where, in particular, a proof of the basic property of proportion is given. It goes like this: “In the correct proportion, the product of the extreme terms is equal to the product of the middle terms. a: b = c: d extreme averages

5 slide

Slide description:

Types of proportions In mathematics, there are two types of proportions: Random (for example, the ratio or proportion between the number of syllables of the longest and shortest names of settlements.) Regular (for example, the proportion between the duration of notes). “Natural” relationships Directly proportional Inversely proportional are widely used in a variety of calculations made by schoolchildren, engineers, administrators, etc. Directly proportional quantities: the length of a circle and its radius; the size of objects and the size of the shadows they cast; Inversely proportional quantities: the duration of the sound of one beat, and the number of beats used in one minute;

6 slide

Slide description:

GOLDEN RATIO In art, the most common proportion is called the “golden ratio”. The golden ratio and even the “divine proportion” were called by ancient and medieval mathematicians the division of a segment in which the length of the entire segment is related to the length of its larger part, as the length of the larger part is to the smaller one. Approximately this ratio is 0.618 ≈5/8. The golden ratio is most often used in works of art, architecture, and is also found in nature.

7 slide

Slide description:

Application of proportion medicine cooking geography Russian language biology physics Fine arts technology Agriculture drawing

8 slide

Slide description:

PROPORTIONS IN THE KITCHEN The concept of proportion is used in cooking. Proportion allows you to calculate the amount of food needed to prepare the same dish for different numbers of guests. Task. To prepare 4 servings of potato casserole, you need to take 0.44 kg of potatoes. How many potatoes do you need to make 10 servings of casserole? Solution 4:0.44=10:x => x=0.44*10:4= 1.1 kg. Answer: you need to take 1 kg 100 g of potatoes.

Slide 9

Slide description:

Proportions and medicine When making medicines, proportions are also observed. Accuracy is necessary here, since if the proportions of the ingredients that make up the medicine are violated, the result may not be medicine, but poison. Task from folk recipes: To prepare propolis tincture, you need to pour crushed propolis with water in a ratio of 2:5. How much water is needed for 150 g of propolis. Solution 2:5=150:x => x=150*5:2= 375g. Answer: you will need 375g. water

10 slide

Slide description:

Proportions in technology lessons The sizes of the elements of a doll's sundress differ from the corresponding sizes of a girl's sundress by the same number of times. Task: The length of the product on the pattern is 75cm. Calculate the scale of the drawing if the length of the sundress is 15 cm. Answer 1:5.

11 slide

Slide description:

Proportion in geography In geography, proportion is also used - scale. Scale is the ratio of the length of a segment on the map to the length of the corresponding segment on the ground. There are different types of scale: numerical, linear and named.

12 slide

Slide description:

Proportions in physics Since ancient times, people have used various levers. Paddle, crowbar, scales, scissors, swing, wheelbarrow, etc. - examples of levers. The gain that the lever gives in the applied effort is determined by the proportion, where M and m are the masses of the loads, and L and l are the “arms” of the lever.

Slide 13

Slide description:

Chemistry Proportion is often used to solve chemistry problems. For example, to find the amount of a substance based on its percentage, it is convenient to use a proportion. Problem: How many kg of salt are in 10 kg of salt water if the percentage of salt is 15%. Solution: 10:100%=X:15%; =>X= 15*10:100=1.5 (kg) salt. Answer: 1.5 kg of salt.

Slide 14

Slide description:

Biology Considering the arrangement of leaves on the common stem of plants, you can notice that between every two pairs of leaves (A and C), the third is located at the golden ratio (point B).

15 slide

Slide description:

Russian language In the Russian language there are proverbs and sayings that establish a direct and inverse relationship. For example: 1) As it comes around, so it will respond. 2) The higher the stump, the higher the shadow. 3) When anger is ahead, the mind is behind. 4) When the pocket is dry, then the court is deaf.

16 slide

Slide description:

Proportion in architecture The proportions of the golden ratio create the impression of harmony and beauty. Therefore, sculptors, architects, and artists have used and continue to use the golden ratio in their works. Golden proportions are present in the dimensions of the facade of the ancient Greek temple of the Parthenon, St. Basil's Cathedral, the Cathedral on the Nerl and many other masterpieces of architecture.

Municipal educational institution "Parfenyevskaya secondary school"

Supervisor Smirnova L.A., mathematics teacher

2010-2011 academic year

Introduction

There are things that cannot be explained. So you come to an empty bench and sit down on it. Where will you sit - in the middle? Or maybe from the very edge? No, most likely, neither one nor the other. You will sit so that the ratio of one part of the bench to the other, relative to your body, will be approximately 1.62. A simple thing, absolutely instinctive... Sitting on a bench, you produced the “golden ratio”. The golden ratio was known back in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the “golden ratio” was studied. Euclid used it when creating his geometry, and Phidias - his immortal sculptures. Plato said that the Universe is arranged according to the “golden ratio”. And Aristotle found a correspondence between the “golden ratio” and the ethical law. The highest harmony of the “golden ratio” will be preached by Leonardo da Vinci and Michelangelo, because beauty and the “golden ratio” are one and the same thing. And Christian mystics will draw pentagrams of the “golden ratio” on the walls of their monasteries, fleeing from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but will never find its exact meaning. An endless series after the decimal point - 1.6180339887... A strange, mysterious, inexplicable thing: this divine proportion mystically accompanies all living things. Inanimate nature does not know what the “golden ratio” is. But you will certainly see this proportion in the curves of sea shells, and in the shape of flowers, and in the appearance of beetles, and in the beautiful human body. Everything living and everything beautiful - everything obeys the divine law, whose name is the “golden ratio”. So what is the “golden ratio”?.. What is this ideal, divine combination? Maybe this is the law of beauty? Or is he still a mystical secret? Scientific phenomenon or ethical principle? The answer is still unknown. More precisely - no, it is known. The “golden ratio” is both, and the third. Only not separately, but simultaneously... And this is his true mystery, his great secret.

The concept of the "golden ratio".

Golden ratio - this is such a proportional division of a segment into unequal parts, in which the smaller segment is related to the larger one, as the larger one is to the whole.

a: b = b: c orc: b = b: a.

This proportion is:

The properties of the golden ratio have created a romantic aura of mystery and almost mystical worship around this number.

For example, in a regular five-pointed star, each segment is divided by a segment intersecting it in the golden ratio (i.e., the ratio of the blue segment to the green, red to blue, green to violet is equal1.618 It is generally accepted that the concept of the golden ratio was introduced into scientific use by Pythagoras. There is an assumption that Pythagoras borrowed his knowledge from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them.

Examples of using the golden ratio

Golden ratio in mathematics

Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden proportion using a compass and ruler. From pointIN a perpendicular equal to half is restoredAB . Received pointWITH connected by a line to a pointA . A segment is plotted on the resulting lineSun ending with a dotD . Line segmentAD transferred to directAB . The resulting pointE divides a segmentAB in the golden ratio ratio. Segments of the golden ratio are expressed as an infinite irrational fractionA.E. = 0.618..., ifAB take as oneBE = 0.382... For practical purposes, approximate values of 0.62 and 0.38 are often used. If the segmentAB taken as 100 parts, then the larger part of the segment is equal to 62, and the smaller part is 38 parts.

The properties of the golden ratio are described by the equation:x 2 – x – 1 = 0. Solution of this equation:

Golden ratio in art

in music

The most extensive study of the manifestations of the golden section in music was undertaken in 1925 by art critic L. Sabaneev. He studied two thousand works by various composers. In his opinion, the temporal extent of a musical work is divided by “certain milestones” that stand out during the perception of music and facilitate the contemplation of the form of the whole. All these musical milestones divide the whole into parts, usually according to the law of the golden ratio.

According to the observations of L. Sabaneev, in the musical works of various composers, not one golden ratio is usually stated, but a whole series of similar sections. Each such section reflects its own musical event, a qualitative leap in the development of the musical theme. In the 1770 works of 42 composers he studied, 3275 golden sections were observed. The number of works in which at least one golden ratio was observed was 1338. The largest number of musical works in which there is a golden ratio are by Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%) , Scriabin (90%), Chopin (92%), Schubert (91%).
All 27 etudes by Chopin have been studied in most detail. 154 golden ratios were discovered in them; in only three studies the golden ratio was absent. In some cases, the structure of a musical work combined symmetry and the golden ratio at the same time; in these cases it was divided into several symmetrical parts, in each of which the golden ratio manifests itself. Beethoven's works are also divided into two symmetrical parts, and within each of them manifestations of the golden proportion are observed.
Moreover, the more talented the composer, the more golden sections are found in his works. In Arensky, Beethoven, Borodin, Haydn, Mozart, Scriabin, Chopin and Schubert, golden sections were found in 90% of all works. According to Sabaneev, the golden ratio leads to the impression of a special harmony of a musical composition. It can be recognized that the golden proportion is a criterion for the harmony of the composition of a musical work.

to the cinema

In cinema, S. Eisenstein artificially constructed the film Battleship Potemkin according to the rules of the “golden ratio”. He broke the tape into five parts. In the first three, the action takes place on a ship. In the last two - in Odessa, where the uprising is unfolding. This transition to the city occurs exactly at the golden ratio point. And each part has its own fracture, which occurs according to the law of the golden ratio.

Golden ratio in painting

Moving on to examples of the “golden ratio” in painting, one cannot help but focus on the work of Leonardo da Vinci.His personality is one of the mysteries of history. Leonardo da Vinci himself said: “Let no one who is not a mathematician dare to read my works.”Let's look closely at the painting "La Gioconda".The portrait of Mona Lisa (La Gioconda) has attracted the attention of researchers for many years, who discovered that the composition of the picture is based on golden triangles, which are parts of a regular star-shaped pentagon.

Also, the proportion of the golden ratio appears in Shishkin’s painting. In this famous painting by I. I. Shishkin, the motifs of the golden ratio are clearly visible. A brightly sunlit pine tree (standing in the foreground) divides the length of the picture according to the golden ratio. To the right of the pine tree is a sunlit hillock. It divides the right side of the picture horizontally according to the golden ratio.

In Raphael's painting "The Massacre of the Innocents" another element of the golden proportion is visible - the golden spiral. In Raphael's preparatory sketch, red lines are drawn running from the semantic center of the composition - the point where the warrior's fingers closed around the child's ankle - along the figures of the child, the woman holding him close, the warrior with his sword raised, and then along the figures of the same group on the right side of the sketch . It is unknown whether Raphael built the golden spiral or felt it.

T. Cook used the golden ratio when analyzing Sandro Botticelli’s painting “The Birth of Venus.”

Golden ratio in architecture

One of the most beautiful works of ancient Greek architecture is the Parthenon (5th century BC).

The figures show a number of patterns associated with the golden ratio. The proportions of the building can be expressed through various powers of the number Ф=0.618...

On the floor plan of the Parthenon you can also see the "golden rectangles"

We can see the golden ratio in the building of Notre Dame Cathedral (Notre Dame de Paris) and in the Pyramid of Cheops:

Not only were the Egyptian pyramids built in accordance with the perfect proportions of the golden ratio, the same phenomenon was found in the Mexican pyramids.

I decided to look at the plans for Parfenyev's churches and see if there was a golden ratio there. The result is an application (multimedia presentation).

Golden ratio in sculpture

The golden proportion was used by many ancient sculptors. The golden proportion of the statue of Apollo Belvedere is known: the height of the depicted person is divided by the umbilical line in the golden section.

Athena Parthenos Olympian Zeus

Back in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it does not matter what format the picture has - horizontal or vertical. There are only four such points; they divide the image size horizontally and vertically in the golden ratio, i.e. they are located at a distance of approximately 3/8 and 5/8 from the corresponding edges of the plane.

Golden ratio in fonts and household items

Golden proportions in parts of the human body

In 1855, the German researcher of the golden ratio, Professor Zeising, published hiswork "Aesthetic Research". Zeising measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law. The division of the body by the navel point is the most important indicator of the golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man.
The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.
Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo Belvedere in the most detail. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, and poetic meters were studied.

I conducted a similar study in 11th grade. The measurement results are shown in the table.Application (multimedia presentation).

Golden ratio in biology and wildlife

Biological studies have shown that, starting with viruses and plants and ending with the human body, the golden proportion is revealed everywhere, characterizing the proportionality and harmony of their structure. The golden ratio is recognized as a universal law of living systems.

Consider a chicory shoot. A shoot has formed from the main stem. The first leaf was located right there. The shoot makes a strong ejection into space, stops, releases a leaf, but this time is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and ejects again.

If the first emission is taken as 100 units, then the second is equal to 62 units, the third is 38, the fourth is 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions.The impulses of its growth gradually decreased in proportion to the golden ratio. It was found that the numerical series of Fibonacci numbers characterizes the structural organization of many living systems. For example, the helical leaf arrangement on a branch forms a fraction (number of revolutions on the stem/number of leaves in a cycle, eg 2/5; 3/8; 5/13), corresponding to the Fibonacci series. The “golden” proportion of five-petal flowers of apple, pear and many other plants is well known. The carriers of the genetic code - DNA and RNA molecules - have a double helix structure; its dimensions almost completely correspond to the numbers of the Fibonacci series. Goethe emphasized nature's tendency toward spirality.

The spider weaves its web in a spiral pattern. A hurricane is spinning like a spiral. A frightened herd of reindeer scatters in a spiral. Goethe called the spiral the “curve of life.” The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc. In many butterflies, the ratio of the sizes of the thoracic and abdominal parts of the body corresponds to the golden ratio. Folding its wings, the moth forms a regular equilateral triangle. But if you spread your wings, you will see the same principle of dividing the body into 2,3,5,8. The dragonfly is also created according to the laws of the golden proportion: the ratio of the lengths of the tail and body is equal to the ratio of the total length to the length of the tail

In a lizard, the length of its tail is related to the length of the rest of the body as 62 to 38. You can notice the golden proportions if you look closely at a bird's egg.

All living things are created in accordance with the proportion of the Golden Section

Some discoveries and theories of modern science,
related to the "golden ratio"

1.Penrose tiles

In ancient science, the “parquet problem” was widely known, which boils down to densely filling a plane with geometric figures of the same type. As is known, such filling can be done usingtriangles , squares Andhexagons . By usingpentagons ( Pentagons ) such filling is impossible.

Parquet problem

Let's take a closer look againregular pentagon also calledPentagon orpentagram , a flat geometric figure based on the "golden ratio".

Regular pentagon or pentagon

As is known, after drawing diagonals in the pentagon, the original pentagon can be represented as a set of three types of geometric figures. In the center there is a new pentagon formed by the intersection points of the diagonals. The rest of the pentagon includes five isosceles triangles, colored yellow, and five isosceles triangles, colored red. Yellow triangles are "golden" because the ratio of the hip to the base is equal to the golden ratio; they have sharp corners of 36 at the apex and sharp corners at 72 at the foundation. Red triangles are also “golden”, since the ratio of the hip to the base is equal to the golden ratio; they have an obtuse angle of 108 at the apex and sharp corners at 36 at the foundation. Now let's connect two yellow triangles and two red triangles with their bases. As a result we get two"golden" rhombus . The first one (yellow) has an acute angle of 36 and an obtuse angle of 144 . We will call the left rhombusthin rhombus, and the right rhombus isthick rhombus.

"Golden" diamonds

The English mathematician and physicist Rogers Penrose used “golden” diamonds to construct “golden” parquet, which was calledPenrose tiles. Penrose tiles are a combination of thick and thin diamonds.

Penrose Tiles

It is important to emphasize thatPenrose tiles have “pentagonal” symmetry or 5th order symmetry, and the ratio of the number of thick rhombuses to thin ones tends to the golden proportion!

2.Quasicrystals

On November 12, 1984, a short paper published in the prestigious journal Physical Review Letters by Israeli physicist Dan Shechtman provided experimental evidence for the existence of a metal alloy with exceptional properties. When studied by electron diffraction methods, this alloy showed all the signs of a crystal. Its diffraction pattern is composed of bright and regularly spaced dots, just like a crystal. However, this picture is characterized by the presence of “icosahedral” or “pentangonal” symmetry, which is strictly prohibited in the crystal for geometric reasons. Such unusual alloys were calledquasicrystals. In less than a year, many other alloys of this type were discovered. There were so many of them that the quasicrystalline state turned out to be much more common than one might imagine.The discovery of quasicrystals is another scientific confirmation that, perhaps, it is the “golden proportion”, which manifests itself both in the world of living nature and in the world of minerals, that is the main proportion of the Universe.

3.Fullerenes

The term "fullerenes"» are called closed molecules of type C 60 , WITH 70 , WITH 76 , WITH 84 , in which all carbon atoms are located on a spherical or spheroidal surface. In these molecules, the carbon atoms are arranged at the vertices of regular hexagons or pentagons that cover the surface of a sphere or spheroid. The central place among fullerenes is occupied by the C molecule 60 , which is characterized by the greatest symmetry and, as a consequence, the greatest stability. This molecule, which resembles the tire of a soccer ball and has the structureregular truncated icosahedron, The carbon atoms are arranged on a spherical surface at the vertices of 20 regular hexagons and 12 regular pentagons, so that each hexagon is bordered by three hexagons and three pentagons, and each pentagon is bordered by hexagons. "Fullerenes" are essentially "man-made" structures arising from fundamental physics research. They were first synthesized in scientists G. Kroto and R. Smalley (who received the Nobel Prize in 1996 for this discovery). But in they were unexpectedly discovered in rocks , that is, fullerenes turned out to be not only “man-made”, but also natural formations. Now fullerenes are being intensively studied in laboratories in different countries, trying to establish the conditions for their formation, structure, properties and possible areas of application.

4. Resonance theory of the solar system

The revolution frequencies of the planets and the differences in revolution frequencies form a spectrum with an interval equal to the golden ratio.

5. Fibonacci resonances of the genetic code

The establishment by science of the now widely known fact of the amazing simplicity of the basic principles of encoding hereditary information in living organisms is one of the most important discoveries of mankind. This simplicity lies in the fact that hereditary information is encoded by texts of three-letter words -triplets orcodons compiled on the basis of an alphabet of four letters - nitrogenous bases A (adenine), C (cytosine), G (guanine), T (thymine). This recording system is essentially the same for the entire vast variety of diverse living organisms and is calledgenetic code.In 1990, French researcher Jean-Claude Perez, who was working at that time as a researcher at IBM, made a very unexpected discovery in the field of genetic coding. He discovered a mathematical law governing the self-organization of basesT, C, A, G inside DNA. He discovered that successive sets of DNA nucleotides are organized into long-range order structures calledRESONANCES . Resonance represents a special proportion that ensures the division of DNA in accordance with the Fibonacci numbers (1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...). For example, the genetic code -insulin chain has the following form:

AT G-TT G-GT C-AAT -CAG-CAC-CTT - T GT -GGT - T CT -CAC-CT C-GTT - GAA-GCT
- TT G-T AC-CTT -GTT - T GC-GGT -GAA-CGT -GGT - TT C-TT C-T AC-ACT -CCT -AAG-
A.C.T

6. Golden proportion in Cantor’s theory of transfinite sets and quantum physics (E-infinity theory)

In recent years, there has been an increased interest in theoretical physics in the “golden section”. The works of the English physicist of Egyptian origin Mohammed El Nashieh show the connection of the “golden ratio” with quantum physics.

Conclusion

The dramatic history of the Golden Ratio, which lasted several thousand years, at the beginning of the 21st century - the “Age of Harmony” - may end with a great triumph for the Golden Ratio. Penrose tiles, the resonant theory of the Solar system (Molchanov, Butusov), quasicrystals (Shekhtman), fullerenes (Croto and Smalley, Nobel Prize 1996) were only the harbingers of this triumph. “Mathematics of Harmony” (Stakhov), Fibonacci and Lucas hyperbolic functions (Stakhov, Tkachenko, Rozin), “Bodnar geometry”, “Law of structural harmony of systems” (Soroko), “E-infinity theory” (El Nashie), Fibonacci matrices and “golden” square matrices (Stakhov) and, finally, “golden” genematrices (Petukhov) - this is not a complete list of modern scientific discoveries based on the Golden Section. These discoveries give reason to suggest that the Golden Ratio is some kind of “metaphysical” knowledge, a “proto-number”, a “universal code of Nature”, which can become the basis for the further development of science, in particular mathematics, theoretical physics, genetics, and computer science.